Self-centering damping column and damping brace

ABSTRACT

A structural column includes a column body having a length L c  and a cross-section having a depth d that is greater than a width b c , with the column body having two end portions at ends of the length of the column body, a first end cap positioned at the first end portion of the column body where the first end cap has a first width greater than the width b c  of the column body and has a first central line of the first width, and a second end cap positioned at the second end portion of the column body where the second end cap has a second width greater than the width b c  of the column body and has a second central line of the second width. Additionally, the first central line and the second central line are off-center from a central line of the width of the column body.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of, and priority to, U.S. Provisional Patent Application No. 62/622,622, filed on Jan. 26, 2018. The entire contents of the foregoing application are hereby incorporated by reference.

BACKGROUND Technical Field

The present disclosure relates to impact damping structures, and more particularly, to self-centering hysteretic damping structures.

Related Art

Conventional seismic design mitigates seismic response through a structural lateral load resisting system. Inelastic ductile behavior is introduced to provide energy dissipation, such as ductile moment-resisting frames, buckling-restrained bracing (BRB) systems, and ductile shear walls. These elasto-plastic (EP) type of passive control systems may be attractive with lower initial cost compared to utilizing active or semi-active control systems. However, such low post-yield stiffness may induce damage under repeated inelastic deformation and may experience significant residual deformations after a strong earthquake. This leads to a considerable increase in cost of post-earthquake repair or replacement of structural members and concerns of occupancy safety under aftershocks. Therefore, the performance-based seismic design approach has included and emphasized residual structural deformation as a fundamental design parameter.

To address the residual deformation issue, significant effort has been taken to develop self-centering seismic resisting systems. Some of the previous designs adopted pre-stressed tendons to introduce a self-centering feature to traditional yielding systems, including post-tensioned precast concrete walls, post-tensioned connections for moment-resisting frames, pre-stressed reinforced concrete columns for bridges, bracing systems with pre-tensioned tendons combined with energy dissipation from a friction, yielding, viscous or visco-elastic (VE) type of supplemental dampers. Structures utilizing rocking type of behavior to achieve seismic control also ensure self-centering behavior through pre-stressed tendons and also need VE or friction dampers for introducing damping into the system. Although self-centering behavior can be achieved with pre-stressed elements, such elements only provide the restoring force to achieve self-centering. Supplemental elements are required to provide the stiffness and damping for a complete self-centering structural member with energy dissipation. Thus, the design and detailing of this type of self-centering system is relatively complex.

Shape memory alloys (SMA) have been considered for the purpose of obtaining self-centering behavior combined with energy dissipation for seismic resisting systems. SMA is a type of smart material that is able to return to its pre-deformed shape under the control of temperature or stress, which can provide the source of actuation for self-centering behavior. Superelastic SMA-based passive seismic control devices are capable of self-centering behavior as well as considerable energy dissipation, and also have favorable fatigue resistance, durability, and reliability. Tests for the device installed in a braced frame exhibited comparable drift reduction to traditional steel braces with the added benefit of self-centering. However, it has been observed that the damping potential of SMA in superelastic form is typically less than 7% equivalent viscous damping, and excessive stain can degrade the damping and re-centering properties of SMA. Some have tested a self-centering bracing system based on SMA but including friction component for extra damping, which demonstrated an appealing seismic response of two braced frames compare to BRB braces. However, main types of SMA contain titanium and therefore costly. The relatively high cost of SMA is still an obstacle to its wide application for the control of large scale civil structures. Accordingly, there is interest in further developing self-centering impact damping structures.

SUMMARY

The present disclosure relates to self-centering hysteretic damping structures.

In accordance with aspects of the present disclosure, a structural column includes a column body having a length L_(c) and a cross-section where the cross-section has a depth d that is greater than a width b_(c), with the column body having two end portions at ends of the length of the column body, a first end cap positioned at a first end portion of the two ends portions of the column body where the first end cap has a first width that is greater than the width b_(c) of the column body and has a first central line of the first width, and a second end cap positioned at a second end portion of the two ends portions of the column body where the second end cap has a second width that is greater than the width b_(c) of the column body and has a second central line of the second width. Additionally, the first central line of the first width and the second central line of the second width are off-center from a central line of the width of the column body.

In various embodiments, the first end cap has a first cap portion having the first width b_(cp) and a second cap portion, wherein the second cap portion tapers from the first width b_(cp) to the width b_(c) of column body.

In various embodiments, the second cap portion has a length of (b_(cp)−b_(c))/2.

In various embodiments, the first cap portion has a length of t_(cp), and the first end cap is positioned at the first end portion of the column body such that the first cap portion extends beyond an end of the column body by t_(cp)/2.

In various embodiments, the second end cap is identical to the first end cap.

In various embodiments, the cross-section of the column body is rectangular, and the column body, the first end cap, and the second end cap together exhibit self-centering and elastic buckling mode jump behavior characterized by a flag-shaped hysteresis loop that relates axial force to axial displacement.

In various embodiments, the elastic buckling mode jump behavior and the flag-shaped hysteresis loop include: a pre-buckling linear phase, a post-primary-buckling fixed-fixed mode stable phase, a post-primary buckling fixed-fixed mode unstable phase, a forward mode jump phase from fixed-fixed mode to pinned-pinned mode, a post-secondary-buckling pinned-pinned mode phase, and a backward mode jump phase from the pinned-pinned mode to the fixed-fixed mode.

In accordance with aspects of the present disclosure, a structural brace includes a plurality of structural columns where at least one structural column of the plurality of structural columns includes a column body having a length L_(c) and a cross-section where the cross-section has a depth d that is greater than a width b_(c), with the column body having two end portions at ends of the length of the column body, a first end cap positioned at a first end portion of the two ends portions of the column body where the first end cap has a first width that is greater than the width b_(c) of the column body and has a first central line of the first width, and a second end cap positioned at a second end portion of the two ends portions of the column body where the second end cap has a second width that is greater than the width b_(c) of the column body and has a second central line of the second width. Additionally, the first central line of the first width and the second central line of the second width are off-center from a central line of the width of the column body.

In various embodiments, the plurality of structural columns includes structural columns of different lengths.

In various embodiments, the structural brace further includes an inner tube and an outer tube, where the inner tube and the outer tube are configured to place the at least one structural column into compression whenever the inner tube and the outer tube are translated relative to each other.

In various embodiments, the inner tube includes a first inner tube end, a second inner tube end, and an inner tube interior space between the first inner tube end and the second inner tube end, the outer tube includes a first outer tube end, a second outer tube end, and an outer tube interior space between the first outer tube end and the second outer tube end, and the inner tube interior space and the outer tube interior space overlap in an overlapping space between the first inner tube end and the second outer tube end.

In various embodiments, the structural brace further includes a first plate and a second plate within the overlapping space, and at least one pre-stressed strand coupling the first plate to the second plate, where the at least one structural column is held between the first plate and the second plate by the at least one pre-stressed strand when the first plate and the second plate are in an unloaded state.

In various embodiments, the first plate and the second plate compress the at least one structural column when the overlapping space decreases by the first inner tube end and the second outer tube end moving closer to each other, and the first plate and the second plate compress the at least one structural column when the overlapping space increases by the first inner tube end and the second outer tube end moving away from each other.

In various embodiments, when the overlapping space decreases, the first plate and the second plate compress the at least one structural column by the first inner tube end forcing the first plate toward the second plate and by the second outer tube end forcing the second plate toward the first plate.

In various embodiments, when the overlapping space increases, the first plate and the second plate compress the at least one structural column by: a first post in the outer tube interior space entering the overlapping space through the first inner tube end and forcing the first plate toward the second plate, and a second post in the inner tube interior space entering the overlapping space through the second outer tube end and forcing the second plate toward the first plate.

In various embodiments, the structural brace further includes a third plate positioned between the first plate and the second plate, where at least one structural column of the plurality of structural columns is coupled to the first plate and to the third plate and is not coupled to the second plate.

In various embodiments, the structural brace further includes a fourth plate positioned between the first plate and the third plate, where at least one structural column of the plurality of structural columns is coupled to the first plate and to the fourth plate and is not coupled to the second plate or the third plate.

In accordance with aspects of the present disclosure, a method is disclosed for providing a structural column exhibiting an elastic buckling mode jump behavior characterized by a target flag-shaped hysteresis loop that relates axial displacement of the structural column to axial force exerted on the structural column. The method includes accessing dimensional parameters for the structural column, where the structural column includes: a column body having a length L_(c) and a cross-section, wherein the cross-section has a depth d that is greater than a width b_(c), with the column body having two end portions at ends of the length of the column body, and having a first central line A_(c) of the width b_(c), and two end caps positioned at the end portions of the column body, with each end cap including an end cap portion having: a width b_(cp) that is greater than the width b_(c) of the column body, a length t_(cp), and a second central line A_(cp) of the width b_(cp). The dimensional parameters include at least two of: the length L_(c) of the structural column, the width b_(c) of the structural column, the depth b_(c) of the structural column, the length t_(cp) of the end cap portion, the width b_(cp) of the end cap portion, a length L_(t) between ends of the two end caps, or a distance e₀ between the first central line A_(c) and the second central line A_(cp). The method further includes accessing at least one structural column requirement from at least one of a target peak axial force of the hysteresis loop, a target energy dissipation per hysteresis loop cycle, or a target elastic behavior of column materials to achieve self-centering behavior, and adjusting at least one of the dimensional parameters based on the at least one structural column requirement.

In various embodiments, adjusting at least one of the dimensional parameters includes adjusting at least one of the following metrics to achieve the at least one structural column requirement: a slenderness defined as L_(c)/b_(c), a cap thickness ratio defined as t_(cp)/L_(t), a cap width ratio defined as b_(cp)/b_(c), an initial eccentricity ratio defined as e₀/L_(t), a depth ratio defined as d/b_(c), and the length L_(t).

Further details and aspects of exemplary embodiments of the present disclosure are described in more detail below with reference to the appended figures.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other aspects and features of the present disclosure will become more apparent in view of the following detailed description when taken in conjunction with the accompanying drawings wherein like reference numerals identify similar or identical elements and:

FIG. 1A is a side view of an exemplary capped column provided in accordance with the present disclosure;

FIG. 1B is a front view of the capped column of FIG. 1A, provided in accordance with the present disclosure;

FIG. 2A is a diagram of an exemplary capped column in the pre-buckling phase;

FIG. 2B is a diagram of an exemplary capped column in the post-primary-buckling phase (fixed-fixed mode);

FIG. 2C is a diagram of an exemplary capped column in the post-secondary-buckling phase (pinned-pinned mode);

FIG. 3 is a graph showing axial displacement vs. mid-span lateral sway of an exemplary capped column;

FIG. 4A is a graph showing axial force-displacement behavior of an exemplary capped column;

FIG. 4B is a graph showing extreme fiber stress-displacement behavior of an exemplary capped column;

FIG. 5A is a diagram of an analytical model for fixed-fixed mode buckling of an exemplary capped column;

FIG. 5B is a diagram of an analytical model for pinned-pinned mode buckling of an exemplary capped column;

FIG. 6A is a collection of graphs showing axial force vs. axial displacement between analytical model prediction and ANSYS simulation results, under varying geometric properties;

FIG. 6B is a collection of graphs showing mid-span lateral sway vs. maximum extreme fiber stress between analytical model prediction and ANSYS simulation results, under varying geometric properties;

FIG. 7A is a collection of graphs showing effect of varying geometric properties on peak axial force;

FIG. 7B is a collection of graphs showing effect of varying geometric properties on energy dissipation per cycle;

FIG. 8 is a collection of graphs showing effect of varying geometric properties on BMJ trigger point and material linear limit deformation;

FIG. 9 is a diagram of a 3-story braced frame model;

FIG. 10 is a schematic diagram of a BMJ brace in accordance with aspects of the present disclosure;

FIG. 11 is a collection of graphs showing seismic response of the structure of FIG. 9 with conventional brace, buckling-restrained brace, and the BMJ brace of FIG. 10, under earthquake ground motions;

FIG. 12 is a collection of graphs showing axial force and deformation of BRB and BMJ braces under earthquake ground motion LA18 of FIG. 11; and

FIG. 13 is a flow diagram of exemplary operations for configuring and providing a capped column.

DETAILED DESCRIPTION

The present disclosure relates to self-centering hysteretic damping structures. As will be explained below and in connection with the figures, the present disclosure provides a capped structural column and a structural brace that includes such a capped structural column. The present disclosure also includes a method of dimensioning such a capped structural column.

Particular self-centering systems possess a flag-shaped hysteresis loop for passive energy dissipation and response reduction. One such system provides an economical source of passive flag-shaped hysteresis damping through a special two-phase buckling behavior of a press-fit flat-ended cylindrical column. The flag-shaped energy dissipation comes from a shift in the elastic buckling mode of the column. Such post-buckling behavior is enabled by the tilting of the press-fit column flat-ends from full-area contact (i.e., primary buckling with fixed-fixed boundary) to edge contact (i.e., secondary buckling with nominal pined-pinned boundary) under compression. This behavior, denoted as buckling mode jump (BMJ), induces hysteretic damping with a negative slope in stiffness caused by geometric nonlinearity. By appropriately configuring the geometry of the column, material yielding can be avoided and only elastic buckling will occur within the column's working range, which brings damage-free and reusable features.

The combined damping and damage-free features of the BMJ mechanism are attractive for passive seismic design. In accordance with aspects of the present disclosure, large-scale devices capable of undergoing large deformation without yielding are disclosed and are compatible with the expected forces and displacements under seismic loads. In accordance with another aspect, an analytical tool for evaluating the post-buckling behavior (after secondary buckling) and unloading behavior is disclosed.

As described in more detail below, a capped column with a rectangular cross-section is disclosed and is configured to provide desired BMJ behavior for large displacements. Also disclosed is an analytical tool to characterize the full post buckling behavior of the capped column, including both stable and unstable phases, and to guide the design and configuration of a device incorporating the BMJ mechanism. In a further aspect, a self-centering bracing system designed based on BMJ mechanism is disclosed for a braced frame building. As discussed below herein, the seismic responses of the building were compared between the cases of BMJ, BRB, and conventional brace (CB). The comparison shows that the disclosed BMJ brace is able to achieve a comparable seismic response as BRB and avoid the residual drift issues observed in the BRB and CB.

In accordance with aspects of the present disclosure, a configuration with a rectangular cross-section column with a relatively deep depth and end caps can achieve larger restoring forces, a larger flag-shaped loop, larger deformations without material yielding, and improved stability for applications in large-scale civil structures.

A schematic representation of a column 100 with two end caps 120, 130 is shown in FIGS. 1A and 1B. FIG. 1A presents a view in a x-z plane, and FIG. 1B presents a view in y-z plane. The column body 110 of the column 100 has a length L_(c) and has a rectangular or substantially rectangular cross-section with a depth d and a width b_(c). The cross-section may be substantially rectangular but not entirely rectangular due to, for example, manufacturing limitations or imperfections, wear and tear over time, or other factors or limitations, such as slightly rounded corners. The column body 110 has two end portions at the lengthwise ends of the column body. A first end cap 120 is positioned at one of the two end portions, and a second end cap 130 is positioned at the other end portion. Each end cap 120, 130 has a width that is greater than the width b_(c) of the column body. In the illustrated embodiment of FIG. 1A, the end caps 120, 130 are identical and have a central line A_(cp) that bisects the width of each end cap. The column body 110 also has a central line A_(c) that bisects the width of the column body. A shift between the central line A_(c) of the column body 110 and the central line A_(cp) of the end caps 120, 130 is introduced (called an initial eccentricity e₀) to trigger the buckling and control the direction of column deflection under axial load.

In various embodiments, the two end caps 120, 130 need not be identical. In various embodiments, the two end caps may have different widths, but the widths of the end caps are still larger than the width b_(c) of the column body. In various embodiments, each end cap central line can be off-center from the central line of the column body by different distances. In various embodiments, the cross-section of the column body 110 need not be a rectangle and can be another shape.

Referring again to FIG. 1A, each end cap 120, 130 has a portion that has width b_(cp) and a tapered portion that tapers from the width b_(cp) down to the width b_(c) of column body. The portion that has width b_(cp) can have a length of t_(cp). The tapered portion can have a length of (b_(cp)−b_(c))/2. In the illustrated embodiment, the end caps are coupled to the column body such that each end cap extends beyond the end of the column body by a distance of t_(cp)/2.

In various embodiments, the materials of the end caps 120, 130 and the column body 110 can be different. For example, a harder material can be used for the end caps to resist indentation under concentrated stresses during BMJ behavior. The tapered portion of the end cap is also designed to reduce stress concentrations. In various embodiments, the depth d of the column body is designed to be larger than the column width b_(c) to ensure a weak axis about the y-axis and that the buckling occurs in the x-z plane. As a result, the buckling behavior of the proposed capped column can be considered as a 2D behavior in the x-z plane of FIG. 1A.

In accordance with aspects of the present disclosure, the BMJ behavior of the capped column 100 can be analyzed. An example of geometric properties of a capped column 100 is summarized in Table 1. In various embodiments, quasi-static analysis of the exemplary capped column can be performed with finite element program ANSYS Workbench 15.0. The following describes an analysis using the ANSYS tool for a particular embodiment of a capped column 100.

Since buckling only occurs about the weak axis, the ANSYS analysis can be performed in 2D with plane stress elements. The material of the end caps and the column is assigned as structural steel and PMMA (polymethyl methacrylate), respectively. In various embodiments of the ANSYS analysis, only linear elastic material constitutive law is used, as the BMJ behavior is geometrically controlled in elastic buckling behavior. Exemplary properties of the materials are given in Table 2, which shows that PMMA ensures a large modulus of resilience (U_(r)=σ_(y) ²/(2E)) compared to steel for larger elastic deformation capacity of the column, while structural steel guarantees a large indentation hardness compared to PMMA for protection of the end cap from wear. Furthermore, the larger modulus of end caps compared to column enables the end caps to behave nominally as rigid bodies. The PMMA has lower self-weight compared to many civil engineering materials, which may introduce less additional weight to the structure it is applied to and induce less initial deformation due to self-weight.

In various embodiments the ANSYS analysis, to capture the BMJ phenomenon from the change of contact condition of the end cap surface under axial compression, the connections between the surface of end caps and the loading blocks are modeled as rough contacts. This assumption allows for the separation of the two contact surfaces in normal direction but no sliding in transverse direction. In various embodiments of the ANSYS analysis, a pre-strain of 0.1% can be applied to the capped column by the loading blocks to clamp the capped column in place. The pre-strain also brings the capped column slightly closer to its primary buckling load. For simplicity, the connections between end caps and column can be modeled as bonded. In various embodiments of the ANSYS analysis, the mesh size is controlled to be 5 mm for the current geometry scale. The quasi-static analysis can be analyzed using displacement control to capture one full loading-unloading cycle.

TABLE 1 Geometric properties of the sample capped column L_(t) L_(c) b_(c) b_(cp) t_(cp) d e₀ (m) (m) (mm) (mm) (mm) (mm) (mm) 1 0.98 50 60 10 150 0.7

TABLE 2 Material properties of the capped column Indentation Compo- E σ_(y) U_(r) ρ hardness nent Material (MPa) (MPa) (MPa) (g/cm³) (MPa) Column PMMA  3100 120 2.32 1.19 170-190 End cap Structural 200000 250 0.16 7.85  370-2070 steel

FIGS. 2A-2C show the results from the ANSYS analysis. The results suggest that the BMJ behavior through change of end contact condition can be well captured in the quasi-static finite element analysis. In FIG. 2A, the loading block is hidden in the full window for better view of the capped column deformation. In FIGS. 2B and 2C, the loading block is presented in the zoom-in window for clearer observation of the change of boundary condition. From FIG. 2A-2C, the deforming behavior during a full loading-unloading cycle can be overall classified into: pre-buckling phase, post-primary-buckling phase in fixed-fixed mode, and post-secondary-buckling phase in pinned-pinned mode. As shown in the ANSYS analysis, the column 100 maintains the initial shape in the pre-buckling phase (FIG. 2A), while for the post-buckling phases, it holds a fixed-fixed buckled shape (FIG. 2B) before the tilting of end occurs and the capped column jumps into a pinned-pinned buckled shape (FIG. 2C). During the jump, the end cap and loading block contact suddenly changes from surface-to-surface 210 (FIG. 2B) to edge-to-surface 220 (FIG. 2C).

To determine the condition for the onset of the jump in boundary condition, the maximum lateral sway of the capped column, which is located at the mid-span of the column, can be visualized as shown in FIG. 3 for the entire loading-unloading cycle. The negative axis values in FIG. 3 reflects direction of displacement and direction of sway. In FIG. 3, the path of the loop is indicated with arrows along the path. The trigger point of BMJ is highlighted at point (X: −14.1 mm, Y: −55.31 mm) in FIG. 3, which indicates that the change of contact condition occurs approximately when the mid-span lateral sway (55.31 mm) just exceeds the distance between the column central line and the edge of the end cap (b_(cp)/2−e₀+b_(c)/2=54.3 mm). This BMJ trigger condition represents the point that the extreme fiber of the column at mid-span deflects beyond the edge of the end caps along y-axis as shown in FIG. 2B. This analysis indicates that the onset of BMJ from fixed-fixed mode to pinned-pinned mode, which relates to the energy dissipation per cycle, can be controlled by adjusting the width of the end caps.

FIGS. 4A and 4B show analysis of the axial force-displacement and extreme fiber stress-displacement behavior, respectively, of the exemplary capped column under compression (the path of the loop is indicated with arrows along the path). The axial force-displacement relationship in FIG. 4A clearly shows a flag-shaped hysteresis loop. The negative axis values in FIG. 4A reflects direction of displacement and reflects the direction of axial force in the compression direction. Energy dissipation results from the switch between the two different buckling modes. In the post-primary-buckling region, two phases are present (path 2-3 and 3-4 in FIG. 4A), separating the post-primary-buckling phase into a stable and an unstable path, which is discussed further below herein. Considering the point (X: −9.937 mm, Y: −25.26 mm) in FIG. 3 and Point 3 (X: −9.937 mm, Y: −199.1 kN) in FIG. 4A, an assumption for the condition of transition Point 3 (post-primary-buckling change from stable to unstable path) can be made: the mid-span lateral sway (25.26 mm in FIG. 3) approximately exceeds half of the column width (25 mm), which indicates that internal axial force at the mid-span of the column extends beyond the section at the end of the column (not including end caps). Verification of the assumptions on transition points is described later herein.

The complete BMJ behavior of the capped column can be summarized in 6 phases based on FIG. 4A as follows.

(1) 1-2: pre-buckling linear phase;

(2) 2-3: post-primary-buckling fixed-fixed mode stable phase;

(3) 3-4: post-primary-buckling fixed-fixed mode unstable phase;

(4) 4-7: forward mode jump phase (fixed-fixed to pinned-pinned);

(5) 5-6: post-secondary-buckling pinned-pinned mode phase;

(6) 6-8: backward mode jump phase (pinned-pinned to fixed-fixed).

Referring to FIG. 4B, during BMJ behavior, the maximum stress in the capped column can be monitored to confirm elastic material behavior and, therefore, confirm that a reusable feature can be obtained. The BMJ mechanism should supply elastic-only buckling to maintain the benefits of damage-free self-centering and hysteretic damping. Thus, the energy dissipation is generated from the geometric nonlinearity, and the analysis can confirm that material yielding can be avoided, especially in the column itself. The maximum extreme fiber stress in the column is considered as a metric for maintaining elastic material behavior. The extreme fiber stress can be controlled through the design of the geometry of the capped column, which is discussed later herein. Based on FIG. 4B, the analysis for the sample capped column confirms that the column extreme fiber stress remains below the material yield strength (σ_(y)=120 MPa) in the full loading-unloading cycle.

From the ANSYS analysis described above, the BMJ behavior is observed to be sensitive to the geometric properties of the capped column. In accordance with aspects of the present disclosure, a 2D analytical model can be used to estimate or determine the sensitivity of the BMJ behavior to geometric parameters, which can serve as an aide in customizing a capped column for particular applications. The analytical model is described below for the 6 BMJ phases listed above.

Phase 1: Pre-Buckling Linear Phase

This phase follows the linear axial behavior of the column with no lateral sway. The equations for predicting axial force (P₁) and stress (σ₁) from axial displacement (d₁) in the pre-buckling linear phase (1) are:

$\begin{matrix} {P_{1} = \frac{E_{c}A_{c}d_{1}}{L_{c}}} & (1) \\ {\sigma_{1} = \frac{P_{1}}{A_{c}}} & (2) \end{matrix}$

where E_(c) is the elastic modulus of the column and A_(c) is the cross-section area of the column (without end caps).

Phase 2: Post-Primary-Buckling Fixed-Fixed Mode Stable Phase

This phase starts from the initial elastic buckling of the column with fixed-fixed boundary conditions, which can be predicted by:

$\begin{matrix} {P_{{cr}\; 1} = \frac{\pi^{2}E_{c}I_{c}}{L^{2}}} & (3) \end{matrix}$

where I_(c) is the moment of inertia of the column section, L=0.5L_(c), and the axial displacement can be calculated using d_(cr1)=P_(cr1)L_(c)/(E_(c)A_(c)). Based on symmetry, half of the column, as shown in FIG. 5A, is used to represent the total column behavior. The rotational spring is considered to have infinite stiffness (no rotation) in this phase. The stable fixed-fixed mode post buckling behavior can then be solved with:

$\begin{matrix} {{v_{2}(z)} = {\frac{v_{{ma}\; x}}{2}\left( {1 - {\cos \left( \frac{\pi \; z}{L} \right)}} \right)}} & (4) \\ {P_{2} = {P_{c\; r\; 1}\left( {1 + \frac{{\pi^{2}\left( \frac{v_{{ma}\; x}}{2} \right)}^{2}}{8L^{2}}} \right)}} & (5) \\ {d_{2} = {{2{\int_{0}^{1}{\frac{1}{2}\left( \frac{{dv}_{2}}{dz} \right)^{2}{dz}}}} + {2\frac{\; {P_{2}L}}{E_{c}A_{c}}}}} & (6) \\ {\sigma_{2} = {\frac{E_{c}I_{c}\frac{d^{2}v_{2}}{{dz}^{2}}{_{z = 0}\frac{b_{c}}{2}}}{I_{c}} + \frac{P_{2}}{A_{c}}}} & (7) \end{matrix}$

where v₂ is the deflection function for BMJ Phase 2 of the half column shown in FIG. 5A, z is the axial coordinate, v_(max)=v₂ (L), P₂ is the axial force, and σ₂ is the maximum extreme fiber stress (at z=0) for BMJ Phase 2. From the assumptions mentioned earlier herein, the BMJ Phase 2 ends when it reaches v_(max)=b_(c)/2.

Therefore, the peak axial force that occurs at the end of BMJ Phase 2 (Point 3 in FIG. 4A), can be calculated as:

$\begin{matrix} {P_{peak} = {P_{{cr}\; 1}\left( {1 + \frac{{\pi^{2}\left( \frac{b_{c}}{4} \right)}^{2}}{8L^{2}}} \right)}} & (8) \end{matrix}$

Phase 3: Post-Primary-Buckling Fixed-Fixed Mode Unstable Phase

This phase can be considered as a gradual transition from fixed-fixed mode to pinned-pinned mode. As shown in FIG. 5A, the capped column is simplified using symmetry. The boundary condition in this phase is a varying state between fully fixed and nominally pinned. The end cap is considered as a rigid bar due to the larger relative elastic modulus and smaller relative length. The boundary condition is modeled as a rotational spring connecting the rigid bar (end cap) to a roller support so that the end is free to move in the axial direction and also support an internal moment. The magnitude of the moment in the rotational spring is unknown. However, the trigger point of BMJ (Point 4 in FIG. 4A) can be predicted using an assumed trigger condition based on lateral sway (v_(max)=l+b_(c)/2, where l=b_(cp)/2−e₀) combined with the assumption that the moment is small enough to ignored at the transition point from fixed-fixed to pinned-pinned boundary condition. The assumed trigger condition of lateral sway is based on when the axial stress on the cross-section area at the mid span moves out of the end cap area along in the z-axis projected area, which initiates the change of boundary condition and therefore the buckling mode jump behavior. The negligible small moment at the transition point is assumed from the contact area of the support end cap decreasing towards a pinned-like condition when approaching the transition point, which suggests that the carried moment is approaching zero in the meanwhile.

Using the assumed lateral sway condition at the transition point (Point 4), the corresponding axial force and displacement can be determined. First, an assumption of the lateral deflection function is made. Since the buckled shape in this phase is between the buckled shapes under fully fixed and perfectly pinned boundary conditions, the deflection shape is considered as a combination of the buckled shapes under fixed-fixed and pinned-pinned boundary conditions. The Rayleigh-Ritz method is selected for this problem. The mixed buckled shape can be represented with two generalized degrees of freedom (DOF) a₁ and a₂. Two admissible functions f₁(z) and f₂ (z) satisfying compatibility and essential boundary conditions are assigned the represent the deflection under changing mixed boundary conditions. The lateral deflection function of the capped half-column in BMJ Phase 3 can be written as:

$\begin{matrix} {{v_{3}(z)} = {\sum\limits_{i = 1}^{2}{a_{i}f_{i}}}} & (9) \end{matrix}$ where:

$\begin{matrix} {f_{1} = {1 - {\cos \left( \frac{\pi \; z}{4L} \right)}}} & (10) \\ {f_{2} = {1 - {\cos \left( \frac{\pi \; z}{2L} \right)}}} & (11) \end{matrix}$

Both f₁ and f₂ satisfy essential boundary conditions:

$\begin{matrix} {{{f_{i}(0)} = 0},{{\frac{{df}_{i}}{dz}(0)} = 0}} & (12) \end{matrix}$

The unknown DOF a₁ and a₂ can be determined through the principle of stationary potential energy (Π_(p3)):

$\begin{matrix} {{\frac{\partial\Pi_{p\; 3}}{\partial a_{i}} = 0},{{{for}\mspace{14mu} i} = 1},2} & (13) \end{matrix}$

where Π_(p3) can be derived by assume the small rotation θ of the rigid end cap in this phase can be ignored:

$\begin{matrix} {{\Pi_{p\; 3} = {{\int_{0}^{L}{\left( {{\frac{1}{2}E_{c}{I_{c}\left( \frac{d^{2}v_{c}}{{dz}^{2}} \right)}^{2}} + {P_{c\; r\; 2}\left( {1 - {\frac{1}{2}\left( \frac{{dv}_{3}}{dz} \right)^{2}}} \right)}} \right){dz}}} - {P_{c\; r\; 2}l\; \frac{{dv}_{3}}{dz}}}}}_{z = L} & (14) \end{matrix}$

where P_(cr2) is the axial force of the capped column at the BMJ trigger Point 4 (FIG. 4A). From Eq. (13), a₁ and a₂ can be solved as functions of P_(cr2) and the deflection function in terms of P_(cr2) can be obtained through Eq. (9). Then, based on the trigger condition assumptions made before, P_(cr2) can be solved using the moment equilibrium at z=0:

$\begin{matrix} {{{{- P_{c\; r\; 2}}\frac{b_{c}}{2}} = {E_{c}I_{c}\frac{d^{2}v_{3}}{{dz}^{2}}}}}_{z = 0} & (15) \end{matrix}$

Then, with P_(cr2) known, the deflection function v₃ can be determined from Eq. (9) and (13) for the trigger point (Point 4) as well. The axial displacement and maximum extreme fiber stress σ_(cr2) of the column at Point 4 can also be solved with:

$\begin{matrix} {d_{c\; r\; 2} = {L_{t} - \left( {{2{\int_{0}^{L}{\frac{1}{2}\left( \frac{{dv}_{3}}{dz} \right)^{2}{dz}}}} - {2\; \frac{P_{c\; r\; 2}L}{E_{c}A_{c}}} + {2t_{cp}}} \right)}} & (16) \\ {\sigma_{c\; r\; 2} = {\frac{{{E_{c}I_{c}\frac{d^{2}v_{3}}{{dz}^{2}}}}_{z = 0}\frac{b_{c}}{2}}{I_{c}} + \frac{P_{c\; r\; 2}}{A_{c}}}} & (17) \end{matrix}$

Because the moment in the rotational spring is unknown in Phase 3, a linear relationship between axial force and displacement is assumed between Point 3 and Point 4 to describe the behavior of this phase (FIG. 4A).

Phase 4: Forward Mode Jump Phase

After the trigger point (Point 4), the buckling mode jumps with a change in boundary condition from fixed-fixed (surface contact) to pinned-pinned (edge contact). This transition would be nearly instantaneous in a physical specimen, resulting in a very sharp slope in the force-displacement relationship. In FIG. 4A, the slope observed between Point 4 and Point 7 is not as sharp, which may be due to the step size of the ANSYS analysis tool. For the present analytical model, however, the forward mode jump phase is described by an instantaneous drop in stiffness.

Phase 5: Post-Secondary-Buckling Pinned-Pinned Mode Phase

Similar as the BMJ Phase 3 (between Points 3-4 in FIG. 4A), this phase (between Points 5-6 in FIG. 4A) can be represented with the deflection function given in Eq. (9) (v₅(z)=v₃(z)). The reason is that the pinned-pinned mode of the capped column does not have a perfectly pinned boundary condition for the column at its joint with end caps, as shown in FIG. 5B. But the boundary condition at the edge of end caps can be considered as fully pinned. So a difference from BMJ phase (3) is that the rotational spring is replaced with pinned connection between the rigid bar and the roller support in this phase (FIG. 5B). Because of the rigid connection between column end and end cap, there is still a small moment present at the center of the column end. Therefore, the buckled shape can be considered as a pinned-pinned dominated mixed buckling condition. Another difference from BMJ Phase (3) is that the rotation 61 of the rigid end cap cannot be neglected anymore and the rigid end cap is forced to be a part of the buckled shape of the whole capped column (in phase (4), L=0.5L_(t) instead). This leads to a different calculation of potential energy (Π_(p5)) for BMJ phase (4):

$\begin{matrix} {\Pi_{p\; 5} = \left. {{\int_{0}^{L}{\left( {{\frac{1}{2}E_{c}{I_{c}\left( \frac{d^{2}v_{5}}{{dz}^{2}} \right)}^{2}} + {P_{5}\left( {1 - {\frac{1}{2}\left( \frac{{dv}_{5}}{dz} \right)^{2}}} \right)}} \right){dz}}} - {P_{5}{l\left( {1 - {\frac{1}{2}\left( \left. \frac{{dv}_{5}}{dz} \right|_{z = L} \right)^{2}}} \right)}\frac{{dv}_{5}}{dz}}} \right|_{z = L}} & (18) \end{matrix}$

The DOF a₁ and a₂ for BMJ Phase (4) can then be calculated with Eqs. (13) and (18) (replacing Π_(p3) with Π_(p5)) in terms of Phase (4) (between Points 4-7 in FIG. 4A) axial force P₄. Based on that and the deflection function, v₅ (z) can be obtained using Eq. (9). The relationship between column deflection the axial force can then be obtained as:

$\begin{matrix} {d_{5} = {L_{t} - \left( {{2{\int_{0}^{L}{\frac{1}{2}\; \left( \frac{{dv}_{5}}{dz} \right)^{2}{dz}}}} - {2\frac{P_{5}L}{E_{c}A_{c}}} + {2l\; {\sin \left( \left. \frac{{dv}_{5}}{dz} \right|_{z = L} \right)}}} \right)}} & (19) \end{matrix}$

Under known applied axial force or displacement, the maximum extreme stress can be solved similarly as Eq. (17):

$\begin{matrix} {\sigma_{5} = {\frac{{{E_{c}I_{c}\frac{d^{2}v_{5}}{{dz}^{2}}}}_{z = 0}\frac{b_{c}}{2}}{I_{c}} + \frac{P_{5}}{A_{c}}}} & (20) \end{matrix}$

With the analytical model derived above for Phase (5) (between Points 5-6 in FIG. 4A), the force-displacement relation can be predicted with a varying slope. The end of this phase is then defined at the point when the slope reaches 90 degrees (Point 6 in FIG. 4A).

Phase 6: Backward Mode Jump Phase

Similar as Phase (4), the backward mode jump phase (between Points 6-8 in FIG. 4A) is defined with a line of 90 degree slope, which starts from the end point of Phase (5) and ends when it returns to the linear Phase (1). This also corresponds to the 90 degree slope definition of the end point of Phase (5) in the analytical model.

In accordance with aspects of the present disclosure, the analytical model for Phases (1) through (6) can be verified. To verify the analytical model, the model can be developed with MATLAB R2014a, and the analytical model's predictions can be compared with the results from quasi-static finite element analysis using ANSYS for capped columns with different geometry properties.

In various embodiments, for the verification and comparison, the geometric properties of the capped column are varied, with the example capped column configuration of Table 1 serving as a baseline. In various embodiments, geometric variations can be implemented with respect to one or more of the following quantities: slenderness (L_(c)/b_(c)), cap thickness ratio (t_(cp)/L_(t)), cap width ratio (b_(cp)/b_(c)), initial eccentricity ratio (e₀/L_(t)), depth ratio (d/b_(c)), and member size (L_(b)), by adjusting b_(c), t_(cp), b_(cp), e₀, d, and L_(t) respectively.

FIG. 6A shows a comparison between predictions using the analytical model disclosed above for Phase (1) through Phase (6) and results of the ANSYS analysis, for axial force and axial displacement, and FIG. 6B shows the comparison for mid-span lateral sway and maximum (mid-span) extreme fiber stress. FIGS. 6A and 6B can be used to verify the accuracy of the analytical model and verify good agreement between the analytical model prediction and the ANSYS result. Based on the comparison, the largest discrepancy is the prediction of the BMJ trigger point (Point 4 of FIG. 4A), which is likely due to the assumption of negligible moment in the rotational spring (FIG. 5A) at this trigger point not being perfectly true. However, it is still a valid assumption for approximation, based on the fairly good agreement between the analytical prediction and the numerical ANSYS results, as shown in FIGS. 6A and 6B. Overall, FIGS. 6A and 6B indicate that the analytical model described above is capable of capturing the full loading-unloading BMJ behavior.

Described above are a structural capped column 100 and an analytical model for characterizing buckling mode jump behavior of the capped column, in accordance with aspects of the present disclosure. The following will now describe customizing and configuring the capped column for particular application. In various embodiments, the capped column can be applied in civil infrastructure, maritime vessels, aircraft, and other structures or vehicles that may experience impact forces. Different applications may have different BMJ behavioral requirements, and the customization and configuration described below can be used to satisfy such requirements.

In accordance with aspects of the present disclosure, and as mentioned earlier herein, the BMJ behavior of the capped column can be controlled by the geometric properties of the capped column. The influence of different geometric properties on the behavior can be determined using the analytical models disclosed above herein for Phase (1) through (6). In various embodiments, peak axial force (P_(MAX)), energy dissipation per cycle (E_(d)), axial deformation (normalized with respect to L_(t)) at BMJ trigger point, and material linear limit are evaluated. The geometric properties that affect BMJ behavior can include slenderness, cap thickness ratio, cap width ratio, initial eccentricity ratio, depth ratio, and/or member size, which are defined earlier herein. The results of the analysis are shown in FIGS. 7A, 7B, and 8.

The peak axial force in the hysteresis loop, as shown in FIG. 7A, provides a beneficial feature for limiting the force transferred by the capped column if it is integrated into a structural system. The peak axial force of capped column can be increased with decreasing slenderness and cap thickness ratio and increasing depth ratio and member length. The cap width ratio and initial eccentricity ratio do not affect the peak axial force.

In terms of energy dissipation per hysteresis cycle, FIG. 7B suggests that to obtain more energy dissipation per cycle, the capped column can be designed with larger cap width ratio, depth ratio, or member length or with a lower slenderness, cap thickness ratio, or initial eccentricity ratio. The most efficient way to increase energy dissipation is to lower the slenderness.

For the design of the capped column, the geometric parameters of the capped column can be configured to ensure that the BMJ behavior is triggered in the targeted axial displacement working range and maintains linear material behavior at the same time. For configuring the capped column in that way, FIG. 8 provides a helpful guidance for finding the desired region for the geometric design of the capped column. As shown in FIG. 8, the desired operational region is below the material yielding limit (to ensure elastic behavior) but above the BMJ trigger point line (to ensure that the BMJ mechanism occurs). Therefore, to ensure that the BMJ is triggered and to avoid material yielding, the slenderness can be at least larger than 18 and the cap width ratio cannot exceed 1.44, for the particular capped column base geometry. For more flexible displacement range after BMJ triggered, which is represented by the distance between the material linear limit and BMJ trigger line in FIG. 8, the capped column can be designed with larger slenderness, cap thickness ratio, or initial eccentricity ratio, or with a smaller cap width ratio. The depth ratio and member length show negligible effect on the desired region.

Accordingly, described above are factors for customizing or configuring a capped column's geometric dimensions for achieving a desired BMJ behavior and/or a target hysteresis loop, including, for example, target axial peak force in the hysteresis loop, target energy dissipation per hysteresis loop cycle, and/or target elastic behavior of column materials to achieve self-centering behavior. It is contemplated that other metrics or requirements can be analyzed in the same manner discussed above herein, and that geometric parameters of the capped column can be adjusted to achieve such metrics or requirements. An exemplary method of providing such a structural column is shown in FIG. 13. At block 1310, the method includes accessing dimensional parameters for a capped column, where the parameters include at least two of: length L_(c) of the column body, width b_(c) of the column body, depth b_(c) of the column body, length t_(cp) of the non-tapered end cap portion, width b_(cp) of the non-tapered end cap portion, length L_(t) between ends of the two end caps, or the initial eccentricity e₀. At block 1320, the method accesses at least one structural column requirement among a target peak axial force of the hysteresis loop, a target energy dissipation per hysteresis loop cycle, and/or a target elastic behavior of column materials to achieve self-centering behavior. At block 1330, the method adjusts one or more of the dimensional parameters based on the structural column requirement, or adjusts one or more of the following metrics to achieve the structural column requirement: a slenderness defined as L_(c)/b_(c), a cap thickness ratio defined as t_(cp)/L_(t), a cap width ratio defined as b_(cp)/b_(c) an initial eccentricity ratio defined as e₀/L_(t), a depth ratio defined as d/b_(c), and/or the length L_(t).

Referring now to FIG. 9, there is shown an application of the capped column in a seismic bracing system 900. FIG. 9 illustrates a BMJ brace system in a 3-story steel frame structure which was originally used for a study of BRB frames. An embodiment of the disclosed brace 910 will be described in connection with FIG. 10. It will be understood that FIG. 9 and FIG. 10 are merely exemplary, and variations are contemplated to be within the scope of the present disclosure. For example, a capped column or brace in accordance with the present disclosure need not be used solely for building structures and can be applied to other structures that experience impact forces, such as marine vessels or aircraft, or other buildings or vehicles. Accordingly, FIGS. 9 and 10 are intended to be illustrative and are not limiting.

With continuing reference to FIG. 9, and in accordance with aspects of the present disclosure, the BMJ brace system 900 can be compared with the BRB system originally designed for the 3-story steel frame structure, with respect to seismic performance. In various embodiments, non-linear time history analysis can be conducted with OpenSees for the seismic analysis. For example, FIG. 11 reflects exemplary analysis results of applying twenty earthquake ground motions (LA01-20) developed with a variation of amplitude and frequency content from fault-parallel and fault-normal orientations of earthquake records. The earthquake ground motions correspond to a 10% chance of exceedance in a 50-year period. In various embodiments, other types of ground motion or other ground motions can be analyzed for the structure of FIG. 9, or for other types of structures.

Referring again to FIG. 9, in the illustrated embodiment, the 3-story prototype structure is designed following FEMA building design criterion with a response factor of 8. For the seismic analysis and comparison, only a single braced bay of the 3-story frame with additional gravity column members is modeled, as shown in FIG. 9. In the illustrated comparison, the gravity column running through full height of the frame model is used to account for the contribution of the columns in the unbraced frame to the lateral stiffness of the structure with a moment of inertia of 44.23×10⁻⁴ m⁴ and plastic modulus 4.75×10⁻³ m³. In various embodiments, the seismic mass of each story in the single bay model can be obtained by dividing the seismic mass of the story by the number of braced bays in each principal direction. Global P-Δ effects are considered based on the seismic mass. To account for the behavior of the gusset plates, all beam column connections are modeled as fixed except for the joints at the roof level (pinned). Beams and columns are modeled with nonlinear beam-column elements by considering nonlinear behavior in bending but linear behavior axially. Beams are then assumed to be inextensible by constraining the horizontal degree of freedoms into one at each floor. That is, in FIG. 9, all nodes at the same story level are constrained in the horizontal direction (the direction of input ground motion) due to a rigid floor assumption. The frame is modeled as fixed at the base. The braces are modeled with pin-ended axial elements for both the BRB case and the BMJ case. The modal damping ratio in the first two modes is selected as 5% using Rayleigh damping.

For the bracing system of FIG. 9, the comparison of FIG. 11 is made between BMJ and BRB brace designs. In various embodiments, an additional comparison to a conventional brace (CB) design can be conducted. For the BRB, the tension capacity and axial stiffness are listed in Table 3, and the compression capacity is assumed to be 1.1 times the tension capacity. The BRB can be modeled with the SteelBRB element in OpenSees.

Referring now to the BMJ brace 910, a schematic configuration is provided in FIG. 10. The structural brace 910 includes an inner tube 920 and an outer tube 930. As shown in FIG. 10, the outer tube 930 has a larger cross-sectional dimension than the cross-sectional dimension of the inner tube 920. A portion of the inner tube 920 is within the outer tube 930, and another portion of the inner tube 920 is outside of the outer tube 930.

Within the inner tube 920 and the outer tube 930 are multiple plates 940, and capped columns 950 according to the present disclosure are positioned between pairs of plates. The inner and outer tubes 920, 930 are configured to place the capped columns 950 into compression for both tension and compression action in the structural brace 910, such that buckling action occurs even under elongation of the brace. The inner and outer tubes 920, 930 can telescope or translate with respect to each other, and slotted end caps maintain compression of the capped columns 950 regardless of tension or compression action of the brace. This operation will now be described in more detail.

With continuing reference to FIG. 10, the inner tube 920 includes an inner tube left end 922, an inner tube right end 924, and an inner tube interior space between the inner tube ends. The outer tube includes an outer tube left end 932, an outer tube right end 934, and an outer tube interior space between the outer tube ends. The interior spaces of the inner and outer tubes overlap, and the overlapping space in FIG. 10 is between the inner tube left end 922 and the outer tube right end 934. Positioned within the overlapping space are a left plate and a right plate and pre-stressed strands 960 coupling the two plates together. The two plates also hold capped columns between them, and the capped columns are held between the plates, when the plates are in an unloaded state, by operation of the pre-stressed strands.

As mentioned above, the left and right plates compress the capped columns when the overlapping space decreases (i.e., the inner tube left end and the outer tube right end move closer to each other), and the two plates also compress the capped columns when the overlapping space increases (i.e., the inner tube left end and the outer tube right end moving away from each other). In particular, in the embodiment of FIG. 10, when the overlapping space decreases, the inner tube left end forces the left plate toward the right plate, and the outer tube right end forces the right plate towards the left plate. When the overlapping space increases, the inner tube left end moves away from the left plate but posts in the interior space of the outer tube move through the inner tube left end and into the overlapping space, thereby forcing the left plate toward the right plate. Additionally, the outer tube right end moves away from the right plate but posts in the interior space of the inner tube move through the outer tube right end and into the overlapping space, thereby forcing the right plate toward the left plate. In this manner, the left and right plates compress the capped columns whenever the inner and outer tubes translate with respect to each other.

In the illustrated embodiment, a number of other plates are positioned between these two plates in a spaced apart arrangement, such that there are gaps between any pair of adjacent plates when the plates are in an unloaded state. In various embodiments, there may be no additional plates between the first two plates. In the illustrated embodiment, pre-stressed strands couple the plates together and maintain the plate positions when the plates are in an unloaded state. Additionally, capped columns are stabilized between each pair of plates by compression provided by the plates due to the pre-stressed strands 960. For sake of clarity, and as shown in FIG. 10, each capped column 950 contacts only two plates 940 and exhibits BMJ behavior when the two plates 940 in contact with the capped column 950 compresses it.

The illustrated design triggers BMJ behavior in both tension and compression using the inner and outer tubes. Hollowed sections of the inner tube ends and outer tube ends enable relative back and forth displacement between the inner and outer tubes. Under brace compressive load, the inner and outer tube are unloaded, the capped column sets which are stabilized in place by the pre-stressed strands are further loaded in compression after closing their individual set gaps, and the pre-stressed strands are still stressed in tension until they are released with the loss of pre-strain. Under brace tensile load, the inner and outer tubes are loaded in tension, and the remainder of the components behaves as before.

To avoid either high forces or accelerations on the structure caused by high stiffness under minor excitations, or abrupt loss of brace stiffness after BMJ occurs, multiple sets of capped columns are used, as shown in FIG. 10. Under such configuration, the stiffness can be gradually increased upon the increase in excitation. In various embodiments, the sets of capped columns are placed inside the inner tube and are stabilized using 2% pre-strain with corresponding sets of pre-stressed strands. The end plates are always in contact with the capped column end caps, but the pre-stressed strands will separate from the end plates when the pre-stress is released under compressive deformation of the capped columns. The strands are released before the BMJ behavior is triggered so that the pre-stressed strands will only affect (slightly increase) the axial stiffness within the 2% pre-strain for each column set. Such behavior will be discussed in more detail later herein. Each capped column set (i.e., all capped columns of similar length) should be placed evenly around the center of the brace device for the stability in the buckling phases. Different sets of capped columns can be placed with an offset in radian location so that the inner space of the brace can be more compact, while avoiding interaction between the buckling deformations of different sets. Furthermore, the longer capped column sets will buckle first. Since the lateral sway of the longer capped column is larger than the shorter capped column due to its larger scale, this provides adequate room for the shorter capped column sets to buckle freely. The gaps enable the sets of capped columns to be triggered at different displacements. Thus, the restoring forces from the inner capped columns is only engaged when contact is made, creating a system of multiple triggered BMJ mechanisms.

Referring also to FIG. 9, for the 3-story frame, 4 sets of capped columns, which are assigned with the same materials as shown in Table 2, are included in the BMJ brace, and the details are summarized in Table 4.

In accordance with aspects of the present disclosure, the BMJ brace can be modeled and analyzed by OpenSees as an axial material model using Microsoft Visual C++. The axial force-displacement behavior in the derived analytical model is converted to the corresponding axial stress-strain behavior for the user developed axial material model, and all 6 BMJ phases can be linearized to construct a multilinear material model for faster computation. The total BMJ brace behavior can be modeled using a superposition of multiple axial brace elements with the multilinear material model of different settings corresponding to the different sizes of each column set.

TABLE 3 Properties of the BRBs in the numerical simulation Tensile strength Axial stiffness Story (kN) (kN/m) 1st 520 1.030 × 10⁵ 2nd 872 1.651 × 10⁵ 3rd 1081 1.905 × 10⁵

The natural frequencies of the 3-story frame with BRB and BMJ brace can be determined from the response of the frame under a band limit white noise. In various embodiments, the frequency content of the white noise is limited within 20 Hz and the noise power is determined by ensuring that the structure response is still linear. Under such conditions, the first and second frequencies are 2.44 Hz and 6.25 Hz for the BRB frame and 1.95 Hz and 5.37 Hz for the BMJ braced frame.

The pre-stressed strands add an additional stiffness onto the initial stiffness of the BMJ braces. As the pre-strain is released and the strands separate from the end caps, the brace stiffness will drop. Such influence on the initial stiffness of each column set can be analyzed and observed in the brace behavior under earthquake excitation in terms of relatively steeper initial slopes compared to that after the pre-strain is released (k₁>k₂, which is illustrated with a zoom-in window in the first subplot of FIG. 12). Also, the stiffness considerably drops after primary buckling for each column set. Therefore, the natural frequencies of the BMJ frame are lower than the BRB case in the linear range from the results under white noise, and also expected to be lower than the BRB case after entering the nonlinear phase based on the observation of response period and brace stiffness (the overall slope in the force-displacement plots) under earthquake excitation from FIG. 12.

TABLE 4 Details of BMJ brace at each story Capped column sets in BMJ brace set 1 (outer) set 2 set 3 set 4 (inner) b_(c)/L_(c) = 25 b_(c)/L_(c) = 22 b_(c)/L_(c) = 20 b_(c)/L_(c) = 20 2 columns 2 columns 2 columns 4 columns Linear Linear Linear Linear L_(t) limit L_(t) limit L_(t) limit L_(t) limit Story (m) d/b_(c) (mm) (m) d/b_(c) (mm) (m) d/b_(c) (mm) (m) d/b_(c) (mm) 1st 1.68 3 123 1.668 4 70 1.65 5 48 1.63 10 47 2nd 1.53 3 112 1.518 4 64 1.5 5 43 1.48 10 43 3rd 1.43 3 105 1.418 4 60 1.4 5 40 1.38 10 40

The seismic response of the 3-story frame with CB, BRB, and BMJ braces under LA01-20 earthquake ground motions are summarized in FIG. 11. Based on FEMA-356, the allowable transient and permanent inter-story drift ratios for the life safety performance level are 2.5% and 1.0% respectively for steel moment resisting frames, and for the immediate occupancy performance level are 0.7% and negligible respectively. The BMJ brace case demonstrates a comparable response in maximum drift ratio compared to the BRB case with maximum drift ratios from all LA01-20 excitations controlled below 2% (satisfying allowable transient drift ratio for life safety performance level), which is considerably reduced from the CB case. For the 0.7% allowable maximum drift ratio corresponding to the immediate occupancy performance level, the BMJ case satisfies the criteria for 5 ground motions, the BRB case satisfies the criteria for 15 ground motions, and the CB violates the criteria in all cases. A significant benefit from the self-centering feature of the BMJ brace is observed in the residual drift ratio comparison. There is negligible residual drift (maximum 0.0196%) for the 3-story frame with BMJ brace under the earthquake suite, which satisfies allowable permanent drift ratio for both life safety performance level and immediate occupancy performance level. However, the residual drift criteria is greatly exceeded for the CB case and non-negligible (immediate occupancy performance level) for the BRB case, which implies potential permanent damage to structural members.

The response of the 3-story frame with BRB and BMJ brace designs under the LA18 ground excitation (which produced the maximum residual drift for BRB frame) can be analyzed in detail in FIG. 12, in which brace refers to the y^(th) brace of x^(th) story. As suggested in FIG. 11, the BRB and BMJ frames exhibit a similar maximum drift level with a noticeably more favorable residual drift in the BMJ frame. The BMJ frame exhibits a slight increase in the peak acceleration response of each floor compared to the BRB frame, which is expected with the improvement in deformation response. However, the overall level of floor absolute acceleration is still similar in the two cases.

Furthermore, FIG. 12 summarizes the axial force-displacement behavior of all 6 BMJ braces of FIG. 9, under LA18 earthquake input for both BRB and BMJ case. It can be seen that the BMJ behavior is triggered gradually through multiple sets of capped columns. The overall stiffness from the BMJ brace system prevents excessive deformation, while the BMJ behavior provides some energy dissipation from each set. The energy dissipation of the BMJ brace is smaller than the BRB brace, which contributes to fewer BMJ simulation cases satisfying the 0.7% allowable maximum drift ratio for immediate occupancy performance level. In various embodiments, such displacement performance may be improved with other sources of energy dissipation such as friction damping introduced to the BMJ brace, though the acceleration response may be increased in the meanwhile depending on the design of the system. In the case of the BRB, the energy comes from the yielding of the brace, contributing to the residual drift of the frame. Overall, the proposed BMJ brace is capable of achieving a desirable maximum drift performance as well as eliminating permanent drift. If the earthquakes were followed by an aftershock, the BMJ brace system would be best suited to mitigate further damage to the structure.

Accordingly, described herein is a capped column that introduces the benefits of the buckling mode jump (BMJ) mechanism to civil infrastructure and other applications and especially with a passive self-centering hysteretic damping brace. The BMJ behavior disclosed herein provides an alternative source of flag-shaped hysteresis damping with a self-centering feature. By allowing the end of the capped column to tilt, the boundary conditions change from fixed-fixed to a nominal pinned-pinned condition under increasing deformation. The change of buckling mode during the transition of boundary conditions generates the flag-shaped hysteresis loop without material yielding.

Also disclosed herein is an analytical model to characterize the BMJ behavior for the capped column geometry, which functions as guidance for customizing and configuring a capped column. The analytical model is verified with numerical analysis results from the finite element software ANSYS.

The present disclosure also includes indications on how geometric properties of the capped column affect the BMJ performance and provides guidance for customizing or configuring capped columns. The metrics or quantities which can be configured include the peak axial force, energy dissipation per cycle, axial displacement for triggering BMJ behavior, and limiting material to remain in the linear region.

Furthermore, a potential application for the BMJ behavior of the proposed capped column for civil structures is disclosed for a 3-story braced frame subject to earthquake loading. A schematic design of the BMJ brace incorporating multiple BMJ mechanisms is disclosed. The seismic performance of the 3-story braced frame is analyzed under 20 earthquake ground motions for the case with BMJ brace, BRB, and CB. The disclosed BMJ braced frame provides significant reduction in seismic response from the CB case, and comparable reductions with the BRB case. Moreover, the residual drift is non-negligible in the BRB case and severe in the CB case. On the other hand, the BMJ brace exhibits remarkable benefits with almost zero residual drift under all 20 earthquake excitations. The results demonstrate the benefits of the BMJ behavior of capped column as an economical alternative with its damage-free and reusable feature for achieving self-centering behavior along with flag-shaped damping.

It should be understood that the foregoing description is only illustrative of the present disclosure. Various alternatives and modifications can be devised by those skilled in the art without departing from the disclosure. Accordingly, the present disclosure is intended to embrace all such alternatives, modifications and variances. The embodiments described with reference to the attached drawing figures are presented only to demonstrate certain examples of the disclosure. The embodiments described and illustrated herein are exemplary, and variations are contemplated to be within the scope of the present disclosure. Various embodiments disclosed herein can be combined in ways not expressly described herein, and such combinations are contemplated to be within the scope of the present disclosure. Other elements, steps, methods, and techniques that are insubstantially different from those described above and/or in the appended claims are also intended to be within the scope of the disclosure. 

What is claimed is:
 1. A structural column comprising: a column body having a length L_(c) and a cross-section, wherein the cross-section has a depth d that is greater than a width b_(c), the column body having two end portions at ends of the length of the column body; a first end cap positioned at a first end portion of the two ends portions of the column body, the first end cap having a first width that is greater than the width b_(c) of the column body and having a first central line of the first width; and a second end cap positioned at a second end portion of the two ends portions of the column body, the second end cap having a second width that is greater than the width b_(c) of the column body and having a second central line of the second width, wherein the first central line of the first width and the second central line of the second width are off-center from a central line of the width of the column body.
 2. The structural column of claim 1, wherein the first end cap has a first cap portion having the first width b_(cp) and a second cap portion, wherein the second cap portion tapers from the first width b_(cp) to the width b_(c) of column body.
 3. The structural column of claim 2, wherein the second cap portion has a length of (b_(cp)−b_(c))/2.
 4. The structural column of claim 3, wherein the first cap portion has a length of t_(cp), wherein the first end cap is positioned at the first end portion of the column body such that the first cap portion extends beyond an end of the column body by t_(cp)/2.
 5. The structural column of claim 4, wherein the second end cap is identical to the first end cap.
 6. The structural column of claim 1, wherein the cross-section of the column body is rectangular, and wherein the column body, the first end cap, and the second end cap together exhibit self-centering and elastic buckling mode jump behavior characterized by a flag-shaped hysteresis loop that relates axial force to axial displacement.
 7. The structural column of claim 6, wherein the elastic buckling mode jump behavior and the flag-shaped hysteresis loop include: a pre-buckling linear phase; a post-primary-buckling fixed-fixed mode stable phase; a post-primary buckling fixed-fixed mode unstable phase; a forward mode jump phase from fixed-fixed mode to pinned-pinned mode; a post-secondary-buckling pinned-pinned mode phase; and a backward mode jump phase from the pinned-pinned mode to the fixed-fixed mode.
 8. A structural brace comprising: a plurality of structural columns, wherein at least one structural column of the plurality of structural columns includes: a column body having a length L_(c) and a substantially rectangular cross-section, wherein the substantially rectangular cross-section has a depth d that is greater than a width b_(c) the column body having two end portions at ends of the length of the column body; a first end cap positioned at a first end portion of the two ends portions of the column body, the first end cap having a first width that is greater than the width b_(c) of the column body and having a first central line of the first width; and a second end cap positioned at a second end portion of the two ends portions of the column body, the second end cap having a second width that is greater than the width b_(c) of the column body and having a second central line of the second width, wherein the first central line of the first width and the second central line of the second width are off-center from a central line of the width of the column body.
 9. The structural brace of claim 8, wherein the plurality of structural columns includes structural columns of different lengths.
 10. The structural brace of claim 8, further comprising an inner tube and an outer tube, wherein the inner tube and the outer tube are configured to place the at least one structural column into compression whenever the inner tube and the outer tube are translated relative to each other.
 11. The structural brace of claim 10, wherein: the inner tube includes a first inner tube end, a second inner tube end, and an inner tube interior space between the first inner tube end and the second inner tube end; the outer tube includes a first outer tube end, a second outer tube end, and an outer tube interior space between the first outer tube end and the second outer tube end; and the inner tube interior space and the outer tube interior space overlap in an overlapping space between the first inner tube end and the second outer tube end.
 12. The structural brace of claim 11, further comprising: a first plate and a second plate within the overlapping space; and at least one pre-stressed strand coupling the first plate to the second plate, wherein the at least one structural column is held between the first plate and the second plate by the at least one pre-stressed strand when the first plate and the second plate are in an unloaded state.
 13. The structural brace of claim 12, wherein the first plate and the second plate compress the at least one structural column when the overlapping space decreases by the first inner tube end and the second outer tube end moving closer to each other, and wherein the first plate and the second plate compress the at least one structural column when the overlapping space increases by the first inner tube end and the second outer tube end moving away from each other.
 14. The structural brace of claim 13, wherein when the overlapping space decreases, the first plate and the second plate compress the at least one structural column by the first inner tube end forcing the first plate toward the second plate and by the second outer tube end forcing the second plate toward the first plate.
 15. The structural brace of claim 13, wherein when the overlapping space increases, the first plate and the second plate compress the at least one structural column by: a first post in the outer tube interior space entering the overlapping space through the first inner tube end and forcing the first plate toward the second plate, and a second post in the inner tube interior space entering the overlapping space through the second outer tube end and forcing the second plate toward the first plate.
 16. The structural brace of claim 12, further comprising: a third plate positioned between the first plate and the second plate, wherein at least one structural column of the plurality of structural columns is coupled to the first plate and to the third plate and is not coupled to the second plate.
 17. The structural brace of claim 16, further comprising: a fourth plate positioned between the first plate and the third plate, wherein at least one structural column of the plurality of structural columns is coupled to the first plate and to the fourth plate and is not coupled to the second plate or the third plate.
 18. A method of providing a structural column exhibiting an elastic buckling mode jump behavior characterized by a target flag-shaped hysteresis loop that relates axial displacement of the structural column to axial force exerted on the structural column, the method comprising: accessing dimensional parameters for the structural column, wherein the structural column includes: a column body having a length L_(c) and a cross-section, wherein the cross-section has a depth d that is greater than a width b_(c), the column body having two end portions at ends of the length of the column body, and having a first central line A_(c) of the width b_(c), and two end caps positioned at the end portions of the column body, each end cap including an end cap portion having: a width b_(cp) that is greater than the width b_(c) of the column body, a length t_(cp), and a second central line A_(cp) of the width b_(cp), and wherein the dimensional parameters include at least two of: the length L_(c) of the structural column, the width b_(c) of the structural column, the depth b_(c) of the structural column, the length t_(cp) of the end cap portion, the width b_(cp) of the end cap portion, a length L_(t) between ends of the two end caps; or a distance e₀ between the first central line A_(c) and the second central line A_(cp); accessing at least one structural column requirement from at least one of a target peak axial force of the hysteresis loop, a target energy dissipation per hysteresis loop cycle, or a target elastic behavior of column materials to achieve self-centering behavior; and adjusting at least one of the dimensional parameters based on the at least one structural column requirement.
 19. The method of claim 18, wherein adjusting at least one of the dimensional parameters includes adjusting at least one of the following metrics to achieve the at least one structural column requirement: a slenderness defined as L_(c)/b_(c), a cap thickness ratio defined as t_(cp)/L_(t), a cap width ratio defined as b_(cp)/b_(c), an initial eccentricity ratio defined as e₀/L_(t), a depth ratio defined as d/b_(c), and the length L_(t). 